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arXiv:hep-th/9302055 · Annalen der Physik

δ‐function perturbations and boundary problems by path integration

Ch. Grosche

Published Jan 1, 1993• 42 citations

Abstract

Abstract A wide class of boundary problems in quantum mechanics is discussed by using path integrals. This includes motion in half‐spaces, radial boxes, rings, and moving boundaries. As a preparation the formalism for the incorporation of δ‐function perturbations is outlined, which includes the discussion of multiple δ‐function perturbations, δ‐function perturbations along perpendicular lines and planes, and moving δ‐function perturbations. The limiting process, where the strength of the δ‐function perturbations gets infinitely repulsive, has the effect of producing impenetrable walls at the locations of the δ‐function perturbations, i.e. a consistent description for boundary problems with Dirichlet boundary‐condition emerges. Several examples illustrate the formalism.

DOI: 10.1002/andp.19935050606

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Abstract A wide class of boundary problems in quantum mechanics is discussed by using path integrals. This includes motion in half‐spaces, radial boxes, rings, and moving boundaries. As a preparation the formalism for the incorporation of δ‐function perturbations is outlined, which includes the discussion of multiple δ‐function perturbations, δ‐function perturbations along perpendicular lines and planes, and moving δ‐function perturbations. The limiting process, where the strength of the δ‐function perturbations gets infinitely repulsive, has the effect of producing impenetrable walls at the locations of the δ‐function perturbations, i.e. a consistent description for boundary problems with Dirichlet boundary‐condition emerges. Several examples illustrate the formalism.

delta '-function perturbations and Neumann boundary conditions are incorporated into the path integral formalism. The starting point is consideration of the path integral representation for the one-dimensional Dirac particle together with a relativistic point interaction. The nonrelativistic limit yields either the usual delta -function or a delta '-function perturbation; making their strengths infinitely repulsive one respectively obtains either Dirichlet or Neumann boundary conditions in the path integral.

#delta#'-function perturbations and Neumann boundary conditions are incorporated into the path integral formalism. The starting point is the consideration of the path integral representation for the one dimensional Dirac particle together with a relativistic point interaction. The non-relativistic limit yields either a usual #delta#-function or a #delta#'-function perturbation; making their strengths infinitely repulsive one obtains Dirichlet, respectively Neumann boundary conditions in the path integral. (orig.)

The path integral for the propagator is expanded into a perturbation series, which can be exactly summed in the case of \ensuremath{\delta}-function perturbations giving a closed expression for the (energy-dependent) Green function. Making the strength of the \ensuremath{\delta}-function perturbation infinite repulsive produces a totally reflecting boundary, hence giving a path integral solution in half spaces in terms of the corresponding Green function. The example of the Wood-Saxon potential serves by an appropriate limiting procedure to obtain the Green function for the step potential and the finite potential well in the half space, respectively.

A proposal is given, of how to implement point interactions and boundary conditions in the path integral. The starting point is a path-integral formulation of a Dirac particle in one dimension. The implementation of a point interaction yields, by means of a perturbation expansion, the corresponding Green function for a relativistic particle. In the non-relativistic limit several cases can be distinguished depending on which of the four possibilities of the point interaction has been chosen. By a proper combination of the various possibilities of implementing the point interaction, the whole range of the four-parameter family of boundary conditions for point interactions can be exploited, in the relativistic case as well as in the non-relativistic limit. In addition, making the strength of the point interaction infinitely repulsive yields boundary conditions at finite points on the real line. In particular, Dirichlet and Neumann boundary conditions emerge. The method is illustrated with some examples.

Abstract The incorporation of two‐ and three‐dimensional Δ‐function perturbations into the path integral formalism is discussed. Contrary to the one‐dimensional case, a regularization prescription is needed due to the divergence of the Green function corresponding to a potential V, G(v) (x,y;E) (x, y ϵ ℝ2, ℝ3) for x→y. The known procedure to define proper self‐adjoint extensions for Hamiltonians with point‐interactions can be exploited to define the incorporation of δ‐function perturbations in the path integral. Several examples illustrate the formalism.